Pricing a Perpetuity-Immediate

What a Perpetuity-Immediate Represents

Some financial arrangements promise the same payment every year with no final payment date. Examples might include a hypothetical fund that makes a fixed annual grant indefinitely or an asset that produces a fixed year-end cash flow forever. To price such a stream, the payment amount and interest rate are not enough by themselves: the timing of the first payment must also be clear.

A perpetuity-immediate is a level stream of payments made at the end of each payment period, continuing without end. In this tutorial, the payment period is one year. If the valuation date is time 0, the payments occur at times 1, 2, 3, and so on. In particular, there is no payment at time 0.

Definition: A level annual perpetuity-immediate consists of payments of amount $R$ at the end of each year, with the first payment one year after the valuation date and one payment at each subsequent year-end forever. Its present value is measured at the valuation date.

The interest rate $i$ is the effective annual interest rate for the year between consecutive payments. The model assumes a constant rate, payments that are certain and level, and no change in payment timing. The present value is finite when $i>0$.

The Pricing Formula

At time 0, a payment $R$ due at the end of year $k$ has present value $Rv^k$, where $v=1/(1+i)$. The whole perpetuity is the sum of the present values of every payment. For a positive effective annual rate, this sum gives the standard pricing formula:

P = Rv + Rv^2 + Rv^3 + \cdots = \frac{R}{i}

Here $P$ is the present value at time 0, $R$ is the payment made at each year-end, and $i$ is expressed as a decimal in the calculation. Thus, an effective annual rate of 6% is entered as $0.06$. The formula gives the value of the entire stream, not the value of one payment.

Key result: For a level annual perpetuity-immediate with payment $R$ at times $1,2,3,\ldots$ and constant effective annual interest rate $i>0$, the present value is $P=R/i$. The first payment is discounted for one full year.

Why the Formula Works

The payments are discounted by successive powers of $v$, so their present values form a geometric series. For the first $n$ payments, the present value is $R(v+v^2+\cdots+v^n)$. The finite geometric-series formula gives $R v(1-v^n)/(1-v)$. Since $i>0$, $0<v<1$, and $v^n$ approaches 0 as $n$ increases. Therefore, the infinite sum is $Rv/(1-v)$. Using $v=1/(1+i)$, we have $1-v=i/(1+i)=iv$, so $Rv/(1-v)=R/i$. This also shows why the first payment's timing matters: the series starts with $Rv$, not $R$.

A Reliable Pricing Method

Before substituting numbers, establish the valuation date, payment frequency, and first payment date. Then confirm that the stated rate applies to that same payment period. For an annual perpetuity-immediate, the first payment must be one year after valuation, and the rate must be an effective annual rate. If those conditions hold, use the steps below.

Locate the valuation date.
Set the date at which the price is required, and identify the payment date one period later.

Confirm the annual payment.
Identify the fixed amount paid at every year-end. If the problem describes multiple level annual commitments, add their payments when they share the same dates.

Use the effective annual rate.
Write the rate as a decimal, such as $0.06$ for 6%, and verify that it is positive.

Calculate and interpret the price.
Divide the annual payment by the rate. State the result as a value at the specified valuation date.

Worked Pricing Examples

Worked Example: Pricing a Perpetual Annual Grant

A hypothetical endowment promises a grant of $4,200 at the end of every year forever. The valuation date is today, the first grant is due exactly one year from today, and the effective annual interest rate is 7%. Find the present value of the grant stream.

Method. This is a level annual perpetuity-immediate: the payment amount is constant, the payments are annual, and the first one occurs at time 1. The rate is given as an effective annual rate, so it matches the period between payments.

Convert 7% to decimal form: $i=0.07$. The annual payment is $R=\$4{,}200$. Apply the perpetuity-immediate formula:

P=\frac{R}{i} =\frac{4{,}200}{0.07} =\frac{420{,}000}{7} =60{,}000

The grant stream has a present value of $60,000.00 at the valuation date. As a check, multiplying the price by the rate gives $60{,}000(0.07)=4{,}200$, the stated annual grant. A common mistake would be to use a formula that includes an extra payment at time 0; the problem specifies that the first grant is one year away.

Worked Example: Testing a Quoted Purchase Price

A hypothetical investment pays $5,000 at each year-end forever, with its first payment one year from the purchase date. The effective annual interest rate is 6.25%. A seller quotes a price of $77,500. Determine the present value of the payments and whether the quoted price equals that value.

Method. Price the specified payment stream at the purchase date, then compare the calculated value with the seller's quote. The cash flows meet the definition of a perpetuity-immediate.

The rate in decimal form is $i=0.0625$, and the annual payment is $R=\$5{,}000$. Therefore:

P=\frac{5{,}000}{0.0625} =\frac{5{,}000}{1/16} =5{,}000(16) =80{,}000

The present value is $80,000.00. The quoted price is $77,500, so it is $2,500 below the calculated present value:

$$ 80{,}000-77{,}500=2{,}500 $$

This comparison is only a price-versus-present-value calculation under the stated interest-rate model; it does not establish that the investment is risk-free or that the stated rate is available in practice. A useful arithmetic check is $80{,}000(0.0625)=5{,}000$. The common mistake to avoid is treating the quoted price as the value without checking it against the promised payment and rate.

Worked Example: Combining Two Annual Perpetual Commitments

A hypothetical community fund is expected to pay $1,800 to one program and $2,700 to another at the end of every year, forever. Both programs receive their first payment one year from the valuation date. The effective annual interest rate is 6%. The fund has $70,000 available. Find the combined present value and determine the amount by which the available funds exceed or fall short of that value.

Method. Since both commitments have the same annual payment dates and continue forever, add the two year-end payments to obtain one level annual payment. Then value that combined perpetuity-immediate.

The total payment at each year-end is:

$$ R=1{,}800+2{,}700=4{,}500 $$

The effective annual rate is $i=0.06$. The present value of both commitments is:

P=\frac{4{,}500}{0.06} =\frac{450{,}000}{6} =75{,}000

The commitments have a combined present value of $75,000.00. Compare this with the available $70,000:

$$ 70{,}000-75{,}000=-5{,}000 $$

The fund falls short by $5,000.00. Check the combined price using the annual interest amount: $75{,}000(0.06)=4{,}500$, exactly the total annual payment. A common mistake is to price one program's payment and overlook the other; both payments occur on the same dates and both must be included.

Common Timing and Rate Errors

Most errors in this calculation come from using the right payment and rate with the wrong interpretation. The formula $P=R/i$ applies to the specified annual perpetuity-immediate, not to every indefinite payment stream.

Counting a payment today. A perpetuity-immediate has no payment at time 0. If the first payment is one year away, the first discounted term is $Rv$, not $R$.
Using the wrong rate format. Enter an effective annual rate as a decimal. For example, 6.25% is $0.0625$, not $6.25$. The rate must apply to the annual period between payments.
Pricing only one payment. The value is not $R$, nor is it the present value of just the first payment $Rv$. It includes the present values of all payments forever.
Ignoring different payment dates. Combining payment amounts is appropriate in the examples above because the payments occur at the same year-ends. Payment streams with different timing cannot be combined by simply adding their stated amounts.

For example, suppose an annual payment of $5,250 is due at each year-end, beginning one year from the valuation date, and the effective annual rate is 7%. The correct calculation is $5{,}250/0.07=\$75{,}000$. An incorrect calculation of $5{,}250(1.07)/0.07=\$80{,}250$ includes an extra payment at time 0. That is not the payment schedule described.

Key takeaway: First confirm the payment dates. For a level annual perpetuity-immediate with its first payment one year after valuation, divide the annual payment by the positive effective annual interest rate to obtain the present value.

Check Your Understanding

For each problem, treat the stated interest rate as a constant effective annual rate and the payments as certain. Give the present value at the stated valuation date.

A hypothetical arts fund pays $2,750 at each year-end forever, with the first payment one year from today. The effective annual interest rate is 5.5%. Find the present value. (Answer: $50,000.00)
A perpetual annual award pays $7,200 at each year-end, beginning one year from the valuation date. The effective annual interest rate is 8%. Find its present value. (Answer: $90,000.00)
A hypothetical research fund pays $4,875 annually at year-end forever, first paying one year from now. The effective annual rate is 6.5%. Calculate the value today. (Answer: $75,000.00)
A community trust makes two perpetual annual payments, $2,100 and $3,150, at the same year-end dates. The first payments are one year from the valuation date, and the effective annual interest rate is 7%. Find the combined present value. (Answer: $75,000.00)
A hypothetical scholarship commitment requires $12,000 at every year-end forever, with its first payment due exactly one year after valuation. At a 4% effective annual interest rate, calculate the present value and compare it with an available balance of $295,000. (Answer: Present value $300,000.00; shortfall $5,000.00)

What You'll Learn